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Magnetohydrodynamic mixed convective slip flow over an inclined porous plate with viscous dissipation and Joule heating
Alexandria Engineering Journal (2015) xxx, xxx–xxx
H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
Magnetohydrodynamic mixed convective slip flow over an inclined porous plate with viscous dissipation and Joule heating S. Das a b c
a,*
, R.N. Jana b, O.D. Makinde
c
Department of Mathematics, University of Gour Banga, Malda 732 103, India Department of Applied Mathematics, Vidyasagar University, Midnapore 721 102, India Faculty of Military Science, Stellenbosch University, Private Bag X2, Saldanha 7395, South Africa
Received 28 October 2014; revised 27 January 2015; accepted 8 March 2015
KEYWORDS Magnetohydrodynamic; Mixed convection; Boundary layer; Slip flow and inclined plate
Abstract The combined effects of viscous dissipation and Joule heating on the momentum and thermal transport for the magnetohydrodynamic flow past an inclined plate in both aiding and opposing buoyancy situations have been carried out. The governing non-linear partial differential equations are transformed into a system of coupled non-linear ordinary differential equations using similarity transformations and then solved numerically using the Runge–Kutta fourth order method with shooting technique. Numerical results are obtained for the fluid velocity, temperature as well as the shear stress and the rate of heat transfer at the plate. The results show that there are significant effects of pertinent parameters on the flow fields. ª 2015 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
1. Introduction Magnetohydrodynamic (MHD) mixed convective flows or combined free and forced convection past a flat plate has been widely studied from both theoretical and experimental standpoints over the past a few decades. MHD mixed convective flows occur in many technological and industrial applications, e.g. solar receivers exposed to wind currents, electronic devices cooled by fans, nuclear reactors cooled during emergency * Corresponding author. Tel.: +91 3222 261171. E-mail addresses: [email protected], [email protected] (S. Das). Peer review under responsibility of Faculty of Engineering, Alexandria University.
shutdown, heat exchangers placed in a low-velocity environment, lubrication purposes, drying technologies, flows in the ocean and in the atmosphere [1,2]. Depending on the forced flow direction, the buoyancy forces may aid (aiding or assisting mixed convection) or oppose (opposing mixed convection) the forced flow, causing an increase or decrease in heat transfer rate [3]. The problem of mixed convection resulting from the flow over a heated vertical plate is of considerable theoretical and practical interest. A detailed review of the subject, including exhaustive lists of references, can be found in the books by Bejan [4], Pop and Ingham [5], Jaluria [6] and Chen and Armaly [7]. References [8–17] are some examples of the recent relevant studies existing in the literature. Mukhopadhyay et al. [18] have presented the MHD combined convective flow past a stretching surface. The mixed convection of a viscous
http://dx.doi.org/10.1016/j.aej.2015.03.003 1110-0168 ª 2015 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: S. Das et al., Magnetohydrodynamic mixed convective slip flow over an inclined porous plate with viscous dissipation and Joule heating, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.03.003
2 dissipating fluid about a vertical flat plate has been studied by Aydin and Kaya [19]. The deviation from interfacial thermodynamic equilibrium will lead to a flow regime where the conventional no-slip wall condition is not valid. According to the value of Knudsen number, the flows can be classified into three categories: continuum flow (Kn < 0:01), slip flow (0:01 6 Kn 6 0:1) and transitional flow (0:1 < Kn < 10) [20]. As the flow deviates away from the continuum limit, the conventional no-slip wall boundary condition fails to accurately model the surface interaction between the fluid and the wall boundary due to the low collision frequency. Slip models have been proposed to ameliorate the prediction of the non-continuum phenomenon near wall boundaries within the framework of the continuum assumption. For large values of Knð> 10Þ, the Navier–Stokes equations are not applicable and the kinetic theory of gases must be employed. In many practical applications, the particle adjacent to a solid surface no longer takes the velocity of the surface. The particle has a finite tangential velocity; it slips along the surface. The flow is called slip-flow and this effect cannot be neglected. Cao and Baker [21] have illustrated the slip effects on a mixed convective flow and heat transfer from a vertical plate. Aziz [22] has presented the hydrodynamic and thermal slip boundary layer flow over a flat plate with constant heat flux boundary condition. The combined effects of Joule heating and viscous dissipation on a magnetohydrodynamic free convective flow past a permeable stretching surface with radiative heat transfer have been determined by Chen [23]. Mukhopadhyay [24] has illustrated the slip effects on a unsteady mixed convective flow and heat transfer past a porous stretching surface. Bhattacharyya et al. [25] have investigated an MHD boundary layer slip flow and heat transfer over a flat plate. Rohni et al. [26] have studied an unsteady mixed convective boundary layer slip flow near the stagnation point on a vertical permeable surface embedded in a porous medium. Bhattacharyya et al. [27] have presented a mixed convective boundary layer slip flow over a vertical plate. The unsteady mixed convective flow from a moving vertical plate in a parallel free stream has been studied by Patil et al. [28]. Ellahi et al. [29] have presented a non-Newtonian MHD fluid flow with slip boundary conditions in porous space. A magnetohydrodynamic peristaltic flow of a Jeffrey fluid in eccentric cylinders has been investigated by Nadeem et al. [30]. The effects of temperature dependent viscosity on an MHD flow of non-Newtonian nanofluid in a pipe have been examined by Ellahi [31]. Zeeshan and Ellahi [32] have presented an MHD slip flow of non-Newtonian fluid in a porous space. Sheikholeslami et al. [33] have studied the Cu-water magneto-nanofluid flow and heat transfer. Ellahi et al. [34] have examined the effects of heat transfer and nonlinear slip on the steady Couette flow. Ellahi [35] has presented the magnetohydrodynamic peristaltic flow of Jeffrey fluid in a rectangular duct through a porous medium. Sheikholeslami et al. [36] have reported the CuO-water nanofluid flow and convective heat transfer considering Lorentz forces. Khana et al. [37] have investigated the effects of heat transfer on a peristaltic motion of Oldroyd fluid in the presence of inclined magnetic field. Akbar et al. [38] have investigated the influence of heat generation and heat flux in peristalsis with interaction of nanoparticles. Sheikholeslami et al. [39] have studied the natural convection of a nanofluid in an enclosure with elliptic inner cylinder. A mixed convective boundary layer flow over a vertical slender cylinder has been presented by Ellahi et al. [40].
S. Das et al. The object of this paper was to investigate the combined effects of viscous dissipation and Joule heating on an MHD mixed convective flow past an inclined porous plate. The viscous and Joule dissipation effects are taken into consideration. The governing equations describing the problem are transformed into a non-linear ordinary differential equations by using similarity transformation. The transformed ordinary differential equations were solved numerically using fourth order Runge–Kutta method with the shooting technique. The effects of pertinent parameters on the fluid velocity and temperature have been shown graphically. 2. Mathematical formulation Consider a mixed convective flow of a viscous incompressible electrically conducting fluid past a porous plate which is inclined from the vertical with an acute angle c measured in the clockwise direction and situated in an otherwise quiescent ambient fluid at temperature T1 . Choose a Cartesian coordinates system with x-axis along the plate and the y-axis is measured normal to the sheet in the outward direction toward the fluid (see Fig. 1(a)). A transverse magnetic field of strength B is applied normal to the plate. The plate coincides with the plane y ¼ 0 and the flow being confined to y > 0. It is assumed that the variation of fluid properties is taken to be negligible except for the essential density variation appearing in the gravitational body force. Ohm’s law is Cowling [41] ~þ ~ ~ ; J~ ¼ r E ð1Þ qB ~ E; ~ J~and r are respectively the velocity vector, the where ~ q; B; magnetic field vector, the electric field vector, the current density vector and the electrical conductivity. It is assumed that the magnetic Reynolds number is very small, so that induced magnetic field can be neglected [41]. This assumption is justified since the magnetic Reynolds number is generally very small for metallic liquid or partially ionized fluid. Liquid metals can be used in a range of applications because they
Figure 1(a)
Geometry of the problem.
Please cite this article in press as: S. Das et al., Magnetohydrodynamic mixed convective slip flow over an inclined porous plate with viscous dissipation and Joule heating, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.03.003
Magnetohydrodynamic mixed convective slip flow
3
are nonflammable, nontoxic and environmentally safe. That is why, liquid metals have number of technical applications in source exchangers, electronic pumps, ambient heat exchangers and also used as a heat engine fluid. Moreover, in nuclear power plants sodium, alloys, lead-bismuth and bismuth are extensively utilized in the heat transfer process. Besides that, mercury play its role as a fluid in high-temperature Rankine cycles and also used in reactors in order to reduce the temperature of the system. For power plants which are exerted at extensively high temperature, sodium is treated as heat-engine fluid. Under the above assumptions and following Chen [23], the governing equations of the conservation of mass, momentum, energy in the presence of magnetic field are @u @v þ ¼ 0; @x @y
ð2Þ
@u @u 1 @p @2u B u þv ¼ þ m 2 þ gb ðT T1 Þ cos c Jz ; ð3Þ @x @y q @x @y q 1 @p 0¼ ; ð4Þ q @y 2 @T @T @2T @u 1 qcp u ¼k 2 þl þv þ J2z ; ð5Þ @x @y @y @y r where u and v are the velocity components along the x and ydirections, respectively, T the temperature of the fluid, l the dynamic viscosity of the fluid, q the fluid density, g the acceleration due to gravity, b the coefficient of thermal expanpffiffiffi sion, cp the specific heat at constant pressure, B ¼ B0 = x is the non-uniform magnetic field applied along the y-axis where B0 a constant. The last term in Eq. (3) characterizes the Lorentz force. The last two terms in Eq. (5) indicate the effects of viscous dissipation and Joule heating respectively. The physical boundary conditions are @u @T u¼L ; v ¼ vw ; T ¼ Tw þ K at y ¼ 0; @y @y u ! U1 ; T ! T1 as y ! 1;
ð6Þ
where Tw ¼ T1 þ Tx0 is the variable temperature of the plate, T0 a constant which measures the rate of increase of temperature along the plate, T1 the free stream temperature assumed constant with Tw > T1 and U1 the uniform free stream velocity. When Tw > T1 , the flow is presented as aiding flow since buoyancy effects have a positive component with the free stream velocity. On the other hand, if Tw < T1 , it is presented as opposing flow as buoyancy effects are in the opposite direction with the free stream velocity. L and K are the velocity and thermal slip factors, respectively and when L ¼ K ¼ 0, the noslip condition is recovered. Since the magnetic field is uniform at infinity, therefore Jx ! 0; Jy ! 0 and Jz ! 0 as y ! 1. This implies that Ex ¼ 0; Ey ¼ 0 and Ez ¼ B U1 . Hence, Ohm law yields Jx ¼ 0; Jy ¼ 0 and Jz ¼ r Bðu U1 Þ. Eq. (4) suggests that the pressure p is a function of x only. On the use of the above assumptions and infinity condition, equations (3) and (5) become @u @u @2u r B2 þv ¼ m 2 þ gb ðT T1 Þ cos c ðu U1 Þ; @x @y @y q 2 @T @T @2T @u qcp u ¼k 2 þl þv þ rBðu U1 Þ2 : @x @y @y @y u
ð7Þ ð8Þ
Figure 1(b) Velocity profile f 0 ðgÞ and shear stress profile f 00 ðgÞ for M2 ¼ 0, k ¼ 0:2; d ¼ 0:2; b ¼ 0:1; Ec ¼ 0 and c ¼ 0.
The continuity Eq. (1) is satisfied by introducing a stream function wðx; yÞ such as u¼
@w @w ; v¼ : @y @x
ð9Þ
The following similarity variables are introduced rffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi U1 T T1 ; w ¼ U1 m x fðgÞ; hðgÞ ¼ g¼y ; mx Tw T1
ð10Þ
where g is the similarity variable, fðgÞ the non-dimensional stream function and hðgÞ the non-dimensional temperature. On the use of (9) and (10) in Eqs. (7) and (8), we obtain the following ordinary differential equations 1 f000 þ ff 00 M2 ðf 0 1Þ þ k h cos c ¼ 0; 2 h i 1 2 00 h þ Prf h0 þ Ec Pr f 00 þ M2 ðf 0 1Þ ¼ 0: 2
ð11Þ ð12Þ
2
where M2 ¼ qrB the magnetic parameter, k ¼ gb UT10 the mixed U1 2
1 convection parameter, Ec ¼ cp ðTUw T the Eckert number and 1Þ
lc
Pr ¼ k p the Prandtl number which measures the ratio of momentum diffusivity to the thermal diffusivity. The prime denotes the differentiation with respect to g. The corresponding boundary conditions are fð0Þ ¼ S; f 0 ð0Þ ¼ d f 00 ð0Þ; hð0Þ ¼ 1 þ b h0 ð0Þ; f 0 ! 1; h ! 0 as g ! 1;
ð13Þ q ffiffiffiffiffi ffi pffiffiffiffiffiffiffiffi where S ¼ 2 U1x m vw is the suction parameter, d ¼ L Um 1x is qffiffiffiffiffiffi the velocity slip parameter, b ¼ K Um 1x the thermal slip
parameter. It is noted that S is positive for suction, but S is negative for blowing at the plate. The slip parameters d and b are different for gaseous and liquid flows, they bear similar physical meanings for both. First, the values of d and b are 1 proportional to x2 and hence they describe the stream-wise location along the plate. Second, the slip parameters control the slip boundary conditions for both flows. The magnitudes of the slip parameter can be used to describe the degree of non-continuum condition at the plate, or in other words, how much the flow deviates from the no-slip condition. For
Please cite this article in press as: S. Das et al., Magnetohydrodynamic mixed convective slip flow over an inclined porous plate with viscous dissipation and Joule heating, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.03.003
4
S. Das et al. 3. Numerical solution The non-linearity of the governing Eqs. (11) and (12) leads to use of numerical method. The transformed non-dimensional governing Eqs. (11) and (12) with boundary conditions (13) are converted into simultaneous first order ordinary differential equations and then solved numerically by fourth order Rung–Kutta method with shooting technique [42]. The resulting higher order ordinary differential equations are reduced to first order differential equations by letting y1 ¼ f; y2 ¼ f 0 ; y3 ¼ f 00 ; y4 ¼ h; y5 ¼ h0 : 2
Figure 2(a) Velocity profile for different M when Pr ¼ 0:72; d ¼ 0:5 and c ¼ p4 for buoyancy aided flow (k > 0).
ð14Þ
Thus, the corresponding higher order non-linear differential equations become y01 ¼ y2 ; y02 ¼ y3 ; 1 y03 ¼ M2 ðy2 1Þ k y4 cos c y1 y3 ; 2 y04 ¼ y5 ; h i 1 y05 ¼ Pry1 y5 EcPr y23 þ M2 ðy2 1Þ2 ; 2
instance, if d ¼ b ¼ 0 indicates that the x-location is far downstream from the leading edge of the plate where the slip effects are negligible. On the other hand, a larger values of the slip parameters indicate that the boundary condition deviates more from the no-slip case. As d and b approach to infinity, the velocity and temperature slips at the plate become infinity large, giving a nearly uniform velocity and temperature distribution with the condition: f00 ð0Þ ¼ 0 and h0 ð0Þ ¼ 0. It is noted that for a very large value of d and b that corresponds to a very small x at the leading edge, the boundary layer assumption is not appropriate, and as a consequence, the boundary-layer equations become inaccurate. Kundsen number (Kn) is a deciding factor, which is a measure of molecular mean free path to characteristic length. When Kundsen number is very small, no slip is observed between the surface and the fluid. However, when Kundsen number lies in the range 103 to 0.1, slip occurs at the surface-fluid interaction. Moreover, if a large value of d and b is due to a Knudsen number greater than 0.1, then the Navier–Stokes equation fails to model the transitional or even free molecule flow regime. For this reason, a discussion regarding large values of d and b could be prone to error in nature. We therefore limit the discussion in this paper to a relatively small range of d and b from 0 to 5 as this will cover the slip flow region.
where b and c are unknown which are to be determined such that the boundary conditions y2 ð1Þ and y4 ð1Þ are satisfied. The shooting method is used to guess b and c by iterations until the boundary conditions are satisfied. The resulting differential equations can be integrated by Runge–Kutta fourth order integration scheme. The accuracy of the assumed missing initial condition is checked by comparing the calculated value of the dependent variable at the terminal point with its given value there. If a difference exists, improved values of the missing initial conditions must be obtained and the process is repeated. The numerical computations are done by MATLAB package. The mesh size is taken as g ¼ 0:01. The numerical computation has been carried out for more refined mesh sizes and the results are found to be independent of the
Figure 2(b) Velocity profile for different M2 when Pr ¼ 0:72; d ¼ 0:5 and c ¼ p4 for buoyancy opposed flow (k < 0).
Figure 3(a) Velocity profile for different d when M2 ¼ 0:5; Pr ¼ 0:72 and c ¼ p4 for buoyancy aided flow (k > 0).
ð15Þ
with the initial conditions: y1 ð0Þ ¼ 0; y2 ð0Þ ¼ d y3 ð0Þ; y3 ð0Þ ¼ b; y4 ð0Þ ¼ c; y4 ð0Þ ¼ 1 þ b y5 ð0Þ;
ð16Þ
Please cite this article in press as: S. Das et al., Magnetohydrodynamic mixed convective slip flow over an inclined porous plate with viscous dissipation and Joule heating, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.03.003
Magnetohydrodynamic mixed convective slip flow mesh size. The process is repeated until we get the results correct up to the desired degree of accuracy 106 . Our problem reduces to the problem studied by Chen [23] when the thermal radiation and free stream velocity were not considered. It should be mentioned that in the absence of magnetic field ðM2 ¼ 0Þ and neglecting the viscous and Joule dissipations and inclination ðc ¼ 0Þ, the relevant results are in agreement with the results reported by Bhattacharyya et al. [27] with slight change of notations. The results obtained in particular cases are compared with the results presented by Bhattacharyya et al. [27]. Fig. 1(b) shows the excellent agreement, which justifies the accuracy of present numerical scheme. 4. Results and discussion In order to gain a clear insight of the physical problem, we have discussed the effects of different values of magnetic parameter M2 , mixed convection parameter k, velocity slip parameter (d), thermal slip parameter b, suction/blowing parameter S, Prandtl number Pr and Eckert number Ec and inclination c on the velocity, temperature, heat transfer rate and shear stress at the plate. M2 ¼ 0 represents the case when there is no applied magnetic field. The mixed convection parameter k represents a measure of the effect of the buoyancy in comparison with that of the inertia of the external forced or free stream flow on the heat and fluid flow. Forced convection is the dominant mode of transport when k ! 0, whereas free convection is the dominant mode when k ! 1. For the heated plate case ðTw > T1 Þ, the upward free convection flow caused by the buoyancy is in the same direction with the external forced convection flow. This case is called buoyancy aided mixed convection flow. For the cooling plate case ðTw < T1 Þ, the buoyancy causes a downward free convective flow which is in the opposite direction to that of upward external forced convection flow, which is called the buoyancy opposed mixed convection flow. The values of Pr are chosen 0.72 which represents air at 20 C temperature and 1 atmospheric pressure. Ec ¼ 0 corresponds to no Joule and viscous heating. The values of magnetic parameter and slip parameters are chosen arbitrarily.
Figure 3(b) Velocity profile for different d when M2 ¼ 0:5; Pr ¼ 0:72 and c ¼ p4 for buoyancy opposed flow (k < 0).
5 The fluid velocity fðgÞ0 increases with an increase in magnetic parameter M2 for both cases of buoyancy aided and opposed flows as shown in Fig. 2(a) and 2(b). An increase in the strength of magnetic parameter M2 , the Lorentz force associated with the magnetic field makes the boundary layer thinner. The magnetic lines of forces move past the plate at the free stream velocity. The fluid which is decelerated by the viscous force, receives a push from the magnetic field which counteracts the viscous effects. Hence the velocity of the fluid increases as the parameter M2 increases. Fig. 3(a) and 3(b) shows that the fluid velocity f 0 ðgÞ increases and as a result the momentum boundary layer thickness decreases with an increase in velocity slip parameter d for buoyancy aided/opposed flows. An increase in slip (in magnitude) more fluid get permits to slip past the plate and fluid experiences less drag and as a result, the flow accelerates near the plate and away from the plate this effect diminishes. There is a deceleration in the fluid velocity f 0 ðgÞ as the inclination c increases for both cases of buoyancy aided and opposed flows as shown in Fig. 4(a) and 4(b). The momentum boundary layer is found to be thickened for increasing values of c. For c ¼ p2, the plate is horizontal and for c ¼ 0 the plate assumes a vertical position. The gravitational effect is minimum for c ¼ p2 and maximum for c ¼ 0. The inclination parameter c arises only
Figure 4(a) Velocity profile for different c when Pr ¼ 0:72; M2 ¼ 0:5 and d ¼ 0:5 for buoyancy aided flow (k > 0).
Figure 4(b) Velocity profile for different c when Pr ¼ 0:72; M2 ¼ 0:5 and d ¼ 0:5 for buoyancy opposed flow(k < 0).
Please cite this article in press as: S. Das et al., Magnetohydrodynamic mixed convective slip flow over an inclined porous plate with viscous dissipation and Joule heating, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.03.003
6 in the buoyancy term k cos c in the momentum Eq. (11). Thus, the fluid velocity is found to be maximized at the vertical position of the plate (c ¼ 0) and minimized for the horizontal position of the plate (c ¼ p2). Fig. 5(a) and 5(b) displays the effects of Prandtl number on the fluid velocity f 0 ðgÞ. The fluid velocity f 0 ðgÞ decreases for increasing values of Prandtl number Pr for buoyancy aided/opposed flows. Physically speaking, the Prandtl number is an important parameter in heat transfer processes as it characterizes the ratio of thicknesses of the viscous and thermal boundary layers. Increasing the value of Pr causes the fluid temperature and its boundary layer thickness to decrease significantly as seen from Fig. 9(a) and 9(b). This decrease in temperature produces a net reduction of the thermal buoyancy effect in the momentum equation which results in less induced flow along the plate and consequently, the fluid velocity decreases. The momentum boundary layer thickness generally increases with increasing values of Pr. Physically, it is true as the Prandtl number describes the ratio between momentum diffusivity and thermal diffusivity and hence controls the relative thickness of the momentum and thermal boundary layers. As Pr increases the viscous forces (momentum diffusivity) dominate the thermal diffusivity and consequently decreases the velocity. Fig. 6 shows that the fluid velocity f 0 ðgÞ increases for increasing values of mixed convection parameter k for
Figure 5(a) Velocity profile for different Pr when M2 ¼ 0:5; d ¼ 0:5 and c ¼ p4 for buoyancy aided flow (k > 0).
Figure 5(b) Velocity profile for different Pr when M2 ¼ 0:5; d ¼ 0:5 and c ¼ p4 for buoyancy opposed flow (k < 0).
S. Das et al. suction/blowing. An increase at k in positive direction results in increasing the fluid velocity due to addition of buoyancy-induced flow onto the external forced convection flow. An increase at k in negative direction results in decreasing velocities due to retarding effect of downward buoyancy-induced flow onto the upward external forced convection flow. For the buoyancy-opposing case (k < 0), the velocity profiles are quite similar to the buoyancy-assisted case (k < 0), this is to say, the velocity profiles increase as k enlarges. The physical explanation is that in the buoyancy-opposing case, the buoyancy force plays a negative effect on the fluid motion in the boundary layer. Fig. 7 illustrates that the fluid velocity f 0 ðgÞ increases for increasing values of suction parameter S for both cases of buoyancy aided and opposed flows. Since the effect of suction is to suck away the fluid near the plate, the momentum boundary layer is reduced due to suction velocity. Consequently the velocity increases. Fig. 8(a) and 8(b) reveals that the fluid temperature hðgÞ decreases with an increase in magnetic parameter M2 for both cases of buoyancy aided and opposed flows. The thermal boundary layer thickness increases for increasing values of M2 . This can be attributed to the influence of Ohmic heating due to magnetic field in the flow. Fig. 9(a) and 9(b) represents the variation of fluid temperature hðgÞ with respect to Prandtl number Pr. The graph depicts that the fluid temperature decreases when the values of Prandtl number Pr increase for both cases of buoyancy aided and opposed flows. This is due to the fact that a higher Prandtl number fluid has relatively low thermal conductivity, which reduces conduction and thereby the thermal boundary layer thickness; and as a result, temperature decreases. Increasing Pr is to increase the heat transfer rate at the surface because the temperature gradient at the surface increases. Fig. 10(a) and 10(b) shows that the fluid temperature hðgÞ increases for increasing values of Eckert number Ec for buoyancy aided/opposed flows. The thermal boundary layer thickness decreases with increasing values of Ec. The viscous dissipation, as a heat generation inside the fluid, increases the bulk fluid temperature. This can be attributed to the additional heating in the flow system due to viscous dissipation. For the case of k ¼ 0, the Eckert number does not have any influence on temperature profile since the momentum and energy equations are not coupled. The fluid temperature hðgÞ decreases for increasing values of
Figure 6 Velocity profile for different k when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and c ¼ p4 in the presence of suction/injection.
Please cite this article in press as: S. Das et al., Magnetohydrodynamic mixed convective slip flow over an inclined porous plate with viscous dissipation and Joule heating, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.03.003
Magnetohydrodynamic mixed convective slip flow
7
Figure 7 Velocity profile for different S when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and c ¼ p4 for buoyancy aided/ opposed flow.
Figure 9(a) Temperature profiles for different Pr when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 for buoyancy aided flow (k > 0).
Figure 8(a) Temperature profiles for different M2 when Pr ¼ 0:72; d ¼ 0:5, b ¼ 0:2 and Ec ¼ 0:1 for buoyancy aided flow (k > 0).
Figure 9(b) Temperature profiles for different Pr when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 for buoyancy opposed flow (k < 0).
Figure 8(b) Temperature profiles for different M2 when Pr ¼ 0:72; d ¼ 0:5, b ¼ 0:2 and Ec ¼ 0:1 for buoyancy opposed flow (k < 0).
Figure 10(a) Temperature profiles for different Ec when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 for buoyancy aided flow (k > 0).
Please cite this article in press as: S. Das et al., Magnetohydrodynamic mixed convective slip flow over an inclined porous plate with viscous dissipation and Joule heating, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.03.003
8
S. Das et al.
Figure 10(b) Temperature profiles for different Ec when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 for buoyancy opposed flow (k < 0).
Figure 12(a) Temperature profiles for different b when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 for buoyancy aided flow (k > 0).
Figure 11(a) Temperature profiles for different d when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 for buoyancy aided flow (k > 0).
Figure 12(b) Temperature profiles for different b when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 for buoyancy opposed flow (k < 0).
Figure 11(b) Temperature profiles for different d when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 for buoyancy opposed flow (k < 0).
Figure 13 Temperature profiles for different k when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 in the presence of suction/injection.
Please cite this article in press as: S. Das et al., Magnetohydrodynamic mixed convective slip flow over an inclined porous plate with viscous dissipation and Joule heating, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.03.003
Magnetohydrodynamic mixed convective slip flow
9
Figure 14 Temperature profiles for different S when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 for buoyancy aided/ opposed flow.
velocity slip parameter d for suction/blowing as shown in Fig. 11(a) and 11(b). The thermal boundary layer thickness increases with increasing values of d. Fig. 12(a) and 12(b) exhibits that the fluid temperature hðgÞ decreases for increasing values of thermal slip parameter b for both cases of buoyancy aided and opposed flows. As the thermal slip parameter b increases, less heat is transferred from the plate to the fluid and hence the temperature decreases for both buoyancy aided and opposed flows. The thermal boundary layer thickness becomes thicker for increasing values of b.
Table 1 The shear stress f 00 ð0Þ and the rate of heat transfer h0 ð0Þ at the plate g ¼ 0. M2
S
k
Pr
Ec
d
b
c
f 00 ð0Þ
h0 ð0Þ
0.5 1 2
0.5 0.5 0.5
0.2 0.2 0.2
0.72 0.72 0.72
0.1 0.1 0.1
0.5 0.5 0.5
0.2 0.2 0.2
p 4 p 4 p 4
0.82450 0.87451 0.95597
0.90292 0.89996 0.89573
0.5 0.5 0.5
0.5 0 0.5
0.72597 0.77541 0.82450
0.77433 0.83771 0.90292
0.77918 0.80187 0.82450
0.90121 0.90207 0.90292
0.82522 0.82484 0.82450
0.85728 0.88146 0.90292
0.82428 0.82450 0.82473
0.92146 0.90292 0.88438
0.82450 0.59019 0.45904
0.90292 0.91717 0.92393
0.82946 0.81975 0.81519
1.10717 0.70725 0.51961
0.83388 0.82450 0.80185
0.90326 0.90292 0.90207
0.2 0 0.2 0.25 0.5 0.72 0 0.1 0.2 0.5 1 1.5 0 0.5 1 0 p 4 p 2
Fig. 13 reveals that the fluid temperature hðgÞ decreases for increasing values of mixed convection parameter k for suction/blowing. Physically, in the process of cooling, the free convection currents are carried away from the plate to the free stream and since the free stream is in the upward direction and thus the free currents induce more fluid velocity to enhance. As a result, the thermal boundary layer thickness increases. Fig. 14 shows that the fluid temperature hðgÞ decreases for increasing values of suction parameter S for both cases of buoyancy aided and opposed flows. The explanation for such behavior is that the fluid is brought closer to the plate surface and reduces the thermal boundary layer thickness. As the distance x from the origin increases, the temperature of the plate decreases which is very clear from the expression of the plate temperature Tw ¼ T1 þ Tx0 as shown in temperature figures. For engineering purposes, one is usually interested in the values of the shear stress (skin friction) and the rate of heat transfer at the plate. The shear stress is an important parameter in the heat transfer studies, since it is directly related to the heat transfer coefficients. The increased shear stress is generally a disadvantage in the technical applications, while the increased heat transfer can be exploited in some applications such as heat exchangers, but should be avoided in other such as gas turbine applications, for instance. The numerical values of the rate of heat transfer h0 ð0Þ and the shear stress f 00 ð0Þ at the plate g ¼ 0 are entered in the Table 1 for several values of M2 ; S; k; Pr; Ec; d; b and c. It is seen from the Table 1 that the rate of heat transfer h0 ð0Þ decreases for increasing values of M2 ; Ec; b; c whereas it decreases for increasing values of S; k; Pr and d. This implies that an increase in Prandtl number is accompanied by an enhancement of the heat transfer rate at the surface of the sheet. The underlying physics behind this can be described as follows. When fluid attains a higher Prandtl number, its thermal conductivity is lowered down and so its heat conduction capacity diminishes. Thereby the thermal boundary layer thickness gets reduced. As a consequence, the heat transfer rate at the surface is increased. The increasing magnetic field also increases the thermal boundary layer thickness and as a result the dimensionless heat transfer rate decreases with an increase in the magnetic field. Buoyancy forces can enhance the rate of heat transfer at the plate when they assist the forced convection, and vice versa. The thermal boundary layer thickness decreases with the suction parameter S which causes an increase in the rate of heat transfer. The explanation for such behavior is that the fluid is brought closer to the surface and reduces the thermal boundary layer thickness. On the other hand, h0 ð0Þ < 0 means the heat transfer takes place from the surface to ambient fluid. The shear stress f 00 ð0Þ increases for increasing values of M2 ; S; k; Ec whereas it decreases for increasing values of Pr; d; b and c. 5. Conclusion In this paper, we examine the combined effects of Joule heating and viscous dissipation on an MHD mixed convective flow past an inclined porous plate. Using similarity transformation, the governing equations are transformed into self-similar equations. The numerical solutions of the self-similar equations are
Please cite this article in press as: S. Das et al., Magnetohydrodynamic mixed convective slip flow over an inclined porous plate with viscous dissipation and Joule heating, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.03.003
10 obtained using shooting technique with the help of forth order Runge–Kutta method. The effects of the pertinent parameters are discussed. Numerical results for temperature and velocity are presented graphically for pertinent parameters. Based on the obtained graphical and tabular results, the following conclusions can be summarized as follows: The fluid velocity and temperature enhance when the strength of magnetic field increases. The fluid velocity temperature accelerates due to increasing thermal buoyancy force. The fluid velocity reduces for increasing values of velocity slip parameter. An increase in velocity slip at the sheet leads to increase the fluid temperature within the boundary layer, while an increase in thermal slip reduces the temperature distribution. The fluid temperature inside the boundary layer reduces when the values of the Prandtl number increase. The slip parameters always lead to thinning of the thermal boundary layer. Joule heating and viscous dissipation have the effect to increase the fluid velocity, and accordingly decrease the rate of heat transfer and shear stress at the plate.
References [1] Y. Jaluria, Natural Convection Heat and Mass Transfer, Pergamon Press, Oxford, 1980. [2] B. Gebhart, Y. Jaluria, R.L. Mahajan, B. Sammakia, Buoyancyinduced Flows and Transport, Hemisphere, 1988. [3] O. Aydn, Aiding and opposing mechanisms of mixed convection in a shear and buoyancy-driven cavity, Int. Commun. Heat Mass Transfer 26 (7) (1999) 1019–1028. [4] A. Bejan, Convection Heat Transfer, Wiley, New York, 1995. [5] I. Pop, D.B. Ingham, Convective Heat Transfer, Pergamon, Amsterdam, 2001. [6] Y. Jaluria, Basic of natural convection, in: S. Kakac, R.K. Shah, W. Aung (Eds.), Handbook of Single-Phase Convective Heat Transfer, Wiley, New York, 1987. [7] T.S. Chen, B.F. Armaly, Mixed convection in external flow, in: S. Kakac, R.K. Shah, W. Aung (Eds.), Handbook of SinglePhase Convective Heat Transfer, Wiley, New York, 1987. [8] N. Afzal, T. Hussain, Mixed convection over a horizontal plate, ASME J. Heat Transfer 106 (1984) 240–241. [9] L.S. Yao, Two-dimensional mixed convection along a flat plate, ASME J. Heat Transfer 190 (1987) 440–445. [10] N.G. Kafoussias, D.A.S. Rees, J.E. Daskalakis, Numerical study of the combined free-forced convective laminar boundary layer flow past a vertical isothermal flat plate with temperature-dependent viscosity, Acta Mech. 127 (1-4) (1998) 39–50. [11] M.A. Hossain, M.S. Munir, Mixed convection flow from a vertical flat plate with temperature dependent viscosity, Int. J. Therm. Sci. 39 (2) (2000) 173–183. [12] H.I. Andersson, Slip flow past a stretching surface, Acta Mech. 158 (2002) 121–125. [13] J.H. Merkin, I. Pop, Mixed convection along a vertical surface: similarity solutions for uniform flow, Fluid Dyn. Res. 30 (2002) 233–250. [14] H.I. Abu-Mulaweh, Measurements of laminar mixed convection flow adjacent to an inclined surface with uniform wall heat flux, Int. J. Therm. Sci. 42 (1) (2003) 57–62.
S. Das et al. [15] C.-H. Chen, Combined heat and mass transfer in MHD free convection from a vertical surface with ohmic heating and viscous dissipation, Int. J. Eng. Sci. 42 (2004) 699–713. [16] R.A. Damesh, Magnetohydrodynamics mixed convection from radiate vertical isothermal surface embedded in a saturated porous media, Trans. ASME, J. Appl. Mech 73 (2006) 54– 59. [17] D. Pal, I.S. Shivakumara, Mixed Convection heat transfer from a vertical heated plate embedded in a sparsely packed porous medium, Int. J. Appl. Mech. Eng. 11 (4) (2006) 929–939. [18] S. Mukhopadhyay, G.C. Layek, R.S.R. Gorla, MHD combined convective flow past a stretching surface, Int. J. Fluid Mech. Res. 34 (3) (2007) 244–257. [19] O. Aydin, A. Kaya, Mixed convection of a viscous dissipating fluid about a vertical flat plate, Appl. Math. Model. 31 (2007) 843–853. [20] M. Gad-el-Hak, The fluid mechanics of microdevices – the freeman scholar lecture, J. Fluids Eng. 121 (1) (1999) 5–33. [21] K. Cao, J. Baker, Slip effects on mixed convective flow and heat transfer from a vertical plate, Int. J. Heat Mass Transfer 52 (2009) 3829–3841. [22] A. Aziz, Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 573–580. [23] C.-H. Chen, Combined effects of Joule heating and viscous dissipation on magnetohydrodynamic flow past a permeable, stretching surface with free convection and radiative heat transfer, ASME J. Heat Transfer 132 (2010) 064503– 064504. [24] S. Mukhopadhyay, Effects of slip on unsteady mixed convective flow and heat transfer past a porous stretching surface, Nucl. Eng. Des. 241 (2011) 2660–2665. [25] K. Bhattacharyya, S. Mukhopadhyay, G.C. Layek, MHD boundary layer slip flow and heat transfer over a flat plate, Chin. Phys. Lett. 28 (2) (2011) 024701. [26] A.M. Rohni, S. Ahmad, I. Pop, J.H. Merkin, Unsteady mixed convection boundary layer flow with suction and temperature slip effects near the stagnation point on a vertical permeable surface embedded in a porous medium, Transp. Porous Media 92 (2012) 1–14. [27] K. Bhattacharyya, S. Mukhopadhyay, G.C. Layek, Similarity solution of mixed convective boundary layer slip flow over a vertical plate, Ain. Shams Eng. J. 4 (2013) 299–305. [28] P.M. Patil, D. Anil Kumar, S. Roy, Unsteady thermal radiation mixed convection flow from a moving vertical plate in a parallel free stream: Effect of Newtonian heating, Int. J. Heat Mass Transfer 62 (2013) 534–540. [29] R. Ellahi, A. Riaz, S. Nadeem, M. Mushtaq, Series solutions of nonlinear partial differential equations with slip boundary conditions for non-Newtonian MHD fluid in porous space, J. Appl. Math. Inform. Sci. 7 (1) (2013) 253–261. [30] S. Nadeem, Arshad Riaz, R. Ellahi, M. Mushtaq, Series solutions of magnetohydrodynamic peristaltic flow of a jeffrey fluid in eccentric cylinders, J. Appl. Math. Inform. Sci. 7 (4) (2013) 1441–1449. [31] R. Ellahi, The effects of MHD and temperature dependent viscosity on the flow of non-Newtonian nanofluid in a pipe: analytical solutions, Appl. Math. Model. 37 (3) (2013) 1451– 1457. [32] A. Zeeshan, R. Ellahi, Series solutions of nonlinear partial differential equations with slip boundary conditions for nonNewtonian MHD fluid in porous space, J. Appl. Math. Inform. Sci. 7 (1) (2013) 253–261. [33] M. Sheikholeslami, M.G. Bandpy, R. Ellahi, M. Hassand, Soheil Soleimanie, CVFEM for MHD effect on Cu-water nanofluid flow and heat transfer, J. Magnet. Magnetic Mater. 349 (2014) 188–200.
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Magnetohydrodynamic mixed convective slip flow [34] R. Ellahi, X. Wang, M. Hameed, Effects of heat transfer and nonlinear slip on the steady flow of couette fluid by means of chebyshev spectral method, Zeitschrift Fur Naturforschung A 69a (2014) 1–8. [35] R. Ellahi, The effects of magnetohydrodynamics on peristaltic flow of jeffrey fluid in a rectangular duct through a porous medium, J. Porous Media 17 (2) (2014) 143–157. [36] M. Sheikholeslami, M.G. Bandpy, R. Ellahi, A. Zeeshan, Simulation of CuO-water nanofluid flow and convective heat transfer considering Lorentz forces, J. Magnet. Magnetic Mater. 369 (2014) 69–80. [37] A.A. Khana, R. Ellahi, M.M. Gulzar, M. Sheikholeslami, Effects of heat transfer on peristaltic motion of Oldroyd fluid in the presence of inclined magnetic field, J. Magnet. Magnetic Mater. 372 (2014) 97–106.
11 [38] N.S. Akbar, M. Raza, R. Ellahi, Influence of heat generation and heat flux in peristalsis with interaction of nanoparticles, Eur. Phys. J.-Plus 129 (2014) 185. [39] M. Sheikholeslami, R. Ellahi, M. Hassan, S. Soleimani, A study of natural convection heat transfer in a nanofluid filled enclosure with elliptic inner cylinder, Int. J. Numer. Methods Heat Fluid Flow 24 (8) (2014) 1906–1927. [40] R. Ellahi, A. Riaz, S. Abbasbandy, T. Hayat, K. Vafai, A study on the mixed convection boundary layer flow and heat transfer over a vertical slender cylinder, Thermal Sci. J. 18 (4) (2014) 1247–1258. [41] T.G. Cowling, Magnetohydrodynamics, Interscience Publisher, Inc, New York, 1957. [42] T.Y. Na, Computational Method in Engineering Boundary Value Problems, Academic Press, New York, 1974.
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H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
Magnetohydrodynamic mixed convective slip flow over an inclined porous plate with viscous dissipation and Joule heating S. Das a b c
a,*
, R.N. Jana b, O.D. Makinde
c
Department of Mathematics, University of Gour Banga, Malda 732 103, India Department of Applied Mathematics, Vidyasagar University, Midnapore 721 102, India Faculty of Military Science, Stellenbosch University, Private Bag X2, Saldanha 7395, South Africa
Received 28 October 2014; revised 27 January 2015; accepted 8 March 2015
KEYWORDS Magnetohydrodynamic; Mixed convection; Boundary layer; Slip flow and inclined plate
Abstract The combined effects of viscous dissipation and Joule heating on the momentum and thermal transport for the magnetohydrodynamic flow past an inclined plate in both aiding and opposing buoyancy situations have been carried out. The governing non-linear partial differential equations are transformed into a system of coupled non-linear ordinary differential equations using similarity transformations and then solved numerically using the Runge–Kutta fourth order method with shooting technique. Numerical results are obtained for the fluid velocity, temperature as well as the shear stress and the rate of heat transfer at the plate. The results show that there are significant effects of pertinent parameters on the flow fields. ª 2015 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
1. Introduction Magnetohydrodynamic (MHD) mixed convective flows or combined free and forced convection past a flat plate has been widely studied from both theoretical and experimental standpoints over the past a few decades. MHD mixed convective flows occur in many technological and industrial applications, e.g. solar receivers exposed to wind currents, electronic devices cooled by fans, nuclear reactors cooled during emergency * Corresponding author. Tel.: +91 3222 261171. E-mail addresses: [email protected], [email protected] (S. Das). Peer review under responsibility of Faculty of Engineering, Alexandria University.
shutdown, heat exchangers placed in a low-velocity environment, lubrication purposes, drying technologies, flows in the ocean and in the atmosphere [1,2]. Depending on the forced flow direction, the buoyancy forces may aid (aiding or assisting mixed convection) or oppose (opposing mixed convection) the forced flow, causing an increase or decrease in heat transfer rate [3]. The problem of mixed convection resulting from the flow over a heated vertical plate is of considerable theoretical and practical interest. A detailed review of the subject, including exhaustive lists of references, can be found in the books by Bejan [4], Pop and Ingham [5], Jaluria [6] and Chen and Armaly [7]. References [8–17] are some examples of the recent relevant studies existing in the literature. Mukhopadhyay et al. [18] have presented the MHD combined convective flow past a stretching surface. The mixed convection of a viscous
http://dx.doi.org/10.1016/j.aej.2015.03.003 1110-0168 ª 2015 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: S. Das et al., Magnetohydrodynamic mixed convective slip flow over an inclined porous plate with viscous dissipation and Joule heating, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.03.003
2 dissipating fluid about a vertical flat plate has been studied by Aydin and Kaya [19]. The deviation from interfacial thermodynamic equilibrium will lead to a flow regime where the conventional no-slip wall condition is not valid. According to the value of Knudsen number, the flows can be classified into three categories: continuum flow (Kn < 0:01), slip flow (0:01 6 Kn 6 0:1) and transitional flow (0:1 < Kn < 10) [20]. As the flow deviates away from the continuum limit, the conventional no-slip wall boundary condition fails to accurately model the surface interaction between the fluid and the wall boundary due to the low collision frequency. Slip models have been proposed to ameliorate the prediction of the non-continuum phenomenon near wall boundaries within the framework of the continuum assumption. For large values of Knð> 10Þ, the Navier–Stokes equations are not applicable and the kinetic theory of gases must be employed. In many practical applications, the particle adjacent to a solid surface no longer takes the velocity of the surface. The particle has a finite tangential velocity; it slips along the surface. The flow is called slip-flow and this effect cannot be neglected. Cao and Baker [21] have illustrated the slip effects on a mixed convective flow and heat transfer from a vertical plate. Aziz [22] has presented the hydrodynamic and thermal slip boundary layer flow over a flat plate with constant heat flux boundary condition. The combined effects of Joule heating and viscous dissipation on a magnetohydrodynamic free convective flow past a permeable stretching surface with radiative heat transfer have been determined by Chen [23]. Mukhopadhyay [24] has illustrated the slip effects on a unsteady mixed convective flow and heat transfer past a porous stretching surface. Bhattacharyya et al. [25] have investigated an MHD boundary layer slip flow and heat transfer over a flat plate. Rohni et al. [26] have studied an unsteady mixed convective boundary layer slip flow near the stagnation point on a vertical permeable surface embedded in a porous medium. Bhattacharyya et al. [27] have presented a mixed convective boundary layer slip flow over a vertical plate. The unsteady mixed convective flow from a moving vertical plate in a parallel free stream has been studied by Patil et al. [28]. Ellahi et al. [29] have presented a non-Newtonian MHD fluid flow with slip boundary conditions in porous space. A magnetohydrodynamic peristaltic flow of a Jeffrey fluid in eccentric cylinders has been investigated by Nadeem et al. [30]. The effects of temperature dependent viscosity on an MHD flow of non-Newtonian nanofluid in a pipe have been examined by Ellahi [31]. Zeeshan and Ellahi [32] have presented an MHD slip flow of non-Newtonian fluid in a porous space. Sheikholeslami et al. [33] have studied the Cu-water magneto-nanofluid flow and heat transfer. Ellahi et al. [34] have examined the effects of heat transfer and nonlinear slip on the steady Couette flow. Ellahi [35] has presented the magnetohydrodynamic peristaltic flow of Jeffrey fluid in a rectangular duct through a porous medium. Sheikholeslami et al. [36] have reported the CuO-water nanofluid flow and convective heat transfer considering Lorentz forces. Khana et al. [37] have investigated the effects of heat transfer on a peristaltic motion of Oldroyd fluid in the presence of inclined magnetic field. Akbar et al. [38] have investigated the influence of heat generation and heat flux in peristalsis with interaction of nanoparticles. Sheikholeslami et al. [39] have studied the natural convection of a nanofluid in an enclosure with elliptic inner cylinder. A mixed convective boundary layer flow over a vertical slender cylinder has been presented by Ellahi et al. [40].
S. Das et al. The object of this paper was to investigate the combined effects of viscous dissipation and Joule heating on an MHD mixed convective flow past an inclined porous plate. The viscous and Joule dissipation effects are taken into consideration. The governing equations describing the problem are transformed into a non-linear ordinary differential equations by using similarity transformation. The transformed ordinary differential equations were solved numerically using fourth order Runge–Kutta method with the shooting technique. The effects of pertinent parameters on the fluid velocity and temperature have been shown graphically. 2. Mathematical formulation Consider a mixed convective flow of a viscous incompressible electrically conducting fluid past a porous plate which is inclined from the vertical with an acute angle c measured in the clockwise direction and situated in an otherwise quiescent ambient fluid at temperature T1 . Choose a Cartesian coordinates system with x-axis along the plate and the y-axis is measured normal to the sheet in the outward direction toward the fluid (see Fig. 1(a)). A transverse magnetic field of strength B is applied normal to the plate. The plate coincides with the plane y ¼ 0 and the flow being confined to y > 0. It is assumed that the variation of fluid properties is taken to be negligible except for the essential density variation appearing in the gravitational body force. Ohm’s law is Cowling [41] ~þ ~ ~ ; J~ ¼ r E ð1Þ qB ~ E; ~ J~and r are respectively the velocity vector, the where ~ q; B; magnetic field vector, the electric field vector, the current density vector and the electrical conductivity. It is assumed that the magnetic Reynolds number is very small, so that induced magnetic field can be neglected [41]. This assumption is justified since the magnetic Reynolds number is generally very small for metallic liquid or partially ionized fluid. Liquid metals can be used in a range of applications because they
Figure 1(a)
Geometry of the problem.
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Magnetohydrodynamic mixed convective slip flow
3
are nonflammable, nontoxic and environmentally safe. That is why, liquid metals have number of technical applications in source exchangers, electronic pumps, ambient heat exchangers and also used as a heat engine fluid. Moreover, in nuclear power plants sodium, alloys, lead-bismuth and bismuth are extensively utilized in the heat transfer process. Besides that, mercury play its role as a fluid in high-temperature Rankine cycles and also used in reactors in order to reduce the temperature of the system. For power plants which are exerted at extensively high temperature, sodium is treated as heat-engine fluid. Under the above assumptions and following Chen [23], the governing equations of the conservation of mass, momentum, energy in the presence of magnetic field are @u @v þ ¼ 0; @x @y
ð2Þ
@u @u 1 @p @2u B u þv ¼ þ m 2 þ gb ðT T1 Þ cos c Jz ; ð3Þ @x @y q @x @y q 1 @p 0¼ ; ð4Þ q @y 2 @T @T @2T @u 1 qcp u ¼k 2 þl þv þ J2z ; ð5Þ @x @y @y @y r where u and v are the velocity components along the x and ydirections, respectively, T the temperature of the fluid, l the dynamic viscosity of the fluid, q the fluid density, g the acceleration due to gravity, b the coefficient of thermal expanpffiffiffi sion, cp the specific heat at constant pressure, B ¼ B0 = x is the non-uniform magnetic field applied along the y-axis where B0 a constant. The last term in Eq. (3) characterizes the Lorentz force. The last two terms in Eq. (5) indicate the effects of viscous dissipation and Joule heating respectively. The physical boundary conditions are @u @T u¼L ; v ¼ vw ; T ¼ Tw þ K at y ¼ 0; @y @y u ! U1 ; T ! T1 as y ! 1;
ð6Þ
where Tw ¼ T1 þ Tx0 is the variable temperature of the plate, T0 a constant which measures the rate of increase of temperature along the plate, T1 the free stream temperature assumed constant with Tw > T1 and U1 the uniform free stream velocity. When Tw > T1 , the flow is presented as aiding flow since buoyancy effects have a positive component with the free stream velocity. On the other hand, if Tw < T1 , it is presented as opposing flow as buoyancy effects are in the opposite direction with the free stream velocity. L and K are the velocity and thermal slip factors, respectively and when L ¼ K ¼ 0, the noslip condition is recovered. Since the magnetic field is uniform at infinity, therefore Jx ! 0; Jy ! 0 and Jz ! 0 as y ! 1. This implies that Ex ¼ 0; Ey ¼ 0 and Ez ¼ B U1 . Hence, Ohm law yields Jx ¼ 0; Jy ¼ 0 and Jz ¼ r Bðu U1 Þ. Eq. (4) suggests that the pressure p is a function of x only. On the use of the above assumptions and infinity condition, equations (3) and (5) become @u @u @2u r B2 þv ¼ m 2 þ gb ðT T1 Þ cos c ðu U1 Þ; @x @y @y q 2 @T @T @2T @u qcp u ¼k 2 þl þv þ rBðu U1 Þ2 : @x @y @y @y u
ð7Þ ð8Þ
Figure 1(b) Velocity profile f 0 ðgÞ and shear stress profile f 00 ðgÞ for M2 ¼ 0, k ¼ 0:2; d ¼ 0:2; b ¼ 0:1; Ec ¼ 0 and c ¼ 0.
The continuity Eq. (1) is satisfied by introducing a stream function wðx; yÞ such as u¼
@w @w ; v¼ : @y @x
ð9Þ
The following similarity variables are introduced rffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi U1 T T1 ; w ¼ U1 m x fðgÞ; hðgÞ ¼ g¼y ; mx Tw T1
ð10Þ
where g is the similarity variable, fðgÞ the non-dimensional stream function and hðgÞ the non-dimensional temperature. On the use of (9) and (10) in Eqs. (7) and (8), we obtain the following ordinary differential equations 1 f000 þ ff 00 M2 ðf 0 1Þ þ k h cos c ¼ 0; 2 h i 1 2 00 h þ Prf h0 þ Ec Pr f 00 þ M2 ðf 0 1Þ ¼ 0: 2
ð11Þ ð12Þ
2
where M2 ¼ qrB the magnetic parameter, k ¼ gb UT10 the mixed U1 2
1 convection parameter, Ec ¼ cp ðTUw T the Eckert number and 1Þ
lc
Pr ¼ k p the Prandtl number which measures the ratio of momentum diffusivity to the thermal diffusivity. The prime denotes the differentiation with respect to g. The corresponding boundary conditions are fð0Þ ¼ S; f 0 ð0Þ ¼ d f 00 ð0Þ; hð0Þ ¼ 1 þ b h0 ð0Þ; f 0 ! 1; h ! 0 as g ! 1;
ð13Þ q ffiffiffiffiffi ffi pffiffiffiffiffiffiffiffi where S ¼ 2 U1x m vw is the suction parameter, d ¼ L Um 1x is qffiffiffiffiffiffi the velocity slip parameter, b ¼ K Um 1x the thermal slip
parameter. It is noted that S is positive for suction, but S is negative for blowing at the plate. The slip parameters d and b are different for gaseous and liquid flows, they bear similar physical meanings for both. First, the values of d and b are 1 proportional to x2 and hence they describe the stream-wise location along the plate. Second, the slip parameters control the slip boundary conditions for both flows. The magnitudes of the slip parameter can be used to describe the degree of non-continuum condition at the plate, or in other words, how much the flow deviates from the no-slip condition. For
Please cite this article in press as: S. Das et al., Magnetohydrodynamic mixed convective slip flow over an inclined porous plate with viscous dissipation and Joule heating, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.03.003
4
S. Das et al. 3. Numerical solution The non-linearity of the governing Eqs. (11) and (12) leads to use of numerical method. The transformed non-dimensional governing Eqs. (11) and (12) with boundary conditions (13) are converted into simultaneous first order ordinary differential equations and then solved numerically by fourth order Rung–Kutta method with shooting technique [42]. The resulting higher order ordinary differential equations are reduced to first order differential equations by letting y1 ¼ f; y2 ¼ f 0 ; y3 ¼ f 00 ; y4 ¼ h; y5 ¼ h0 : 2
Figure 2(a) Velocity profile for different M when Pr ¼ 0:72; d ¼ 0:5 and c ¼ p4 for buoyancy aided flow (k > 0).
ð14Þ
Thus, the corresponding higher order non-linear differential equations become y01 ¼ y2 ; y02 ¼ y3 ; 1 y03 ¼ M2 ðy2 1Þ k y4 cos c y1 y3 ; 2 y04 ¼ y5 ; h i 1 y05 ¼ Pry1 y5 EcPr y23 þ M2 ðy2 1Þ2 ; 2
instance, if d ¼ b ¼ 0 indicates that the x-location is far downstream from the leading edge of the plate where the slip effects are negligible. On the other hand, a larger values of the slip parameters indicate that the boundary condition deviates more from the no-slip case. As d and b approach to infinity, the velocity and temperature slips at the plate become infinity large, giving a nearly uniform velocity and temperature distribution with the condition: f00 ð0Þ ¼ 0 and h0 ð0Þ ¼ 0. It is noted that for a very large value of d and b that corresponds to a very small x at the leading edge, the boundary layer assumption is not appropriate, and as a consequence, the boundary-layer equations become inaccurate. Kundsen number (Kn) is a deciding factor, which is a measure of molecular mean free path to characteristic length. When Kundsen number is very small, no slip is observed between the surface and the fluid. However, when Kundsen number lies in the range 103 to 0.1, slip occurs at the surface-fluid interaction. Moreover, if a large value of d and b is due to a Knudsen number greater than 0.1, then the Navier–Stokes equation fails to model the transitional or even free molecule flow regime. For this reason, a discussion regarding large values of d and b could be prone to error in nature. We therefore limit the discussion in this paper to a relatively small range of d and b from 0 to 5 as this will cover the slip flow region.
where b and c are unknown which are to be determined such that the boundary conditions y2 ð1Þ and y4 ð1Þ are satisfied. The shooting method is used to guess b and c by iterations until the boundary conditions are satisfied. The resulting differential equations can be integrated by Runge–Kutta fourth order integration scheme. The accuracy of the assumed missing initial condition is checked by comparing the calculated value of the dependent variable at the terminal point with its given value there. If a difference exists, improved values of the missing initial conditions must be obtained and the process is repeated. The numerical computations are done by MATLAB package. The mesh size is taken as g ¼ 0:01. The numerical computation has been carried out for more refined mesh sizes and the results are found to be independent of the
Figure 2(b) Velocity profile for different M2 when Pr ¼ 0:72; d ¼ 0:5 and c ¼ p4 for buoyancy opposed flow (k < 0).
Figure 3(a) Velocity profile for different d when M2 ¼ 0:5; Pr ¼ 0:72 and c ¼ p4 for buoyancy aided flow (k > 0).
ð15Þ
with the initial conditions: y1 ð0Þ ¼ 0; y2 ð0Þ ¼ d y3 ð0Þ; y3 ð0Þ ¼ b; y4 ð0Þ ¼ c; y4 ð0Þ ¼ 1 þ b y5 ð0Þ;
ð16Þ
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Magnetohydrodynamic mixed convective slip flow mesh size. The process is repeated until we get the results correct up to the desired degree of accuracy 106 . Our problem reduces to the problem studied by Chen [23] when the thermal radiation and free stream velocity were not considered. It should be mentioned that in the absence of magnetic field ðM2 ¼ 0Þ and neglecting the viscous and Joule dissipations and inclination ðc ¼ 0Þ, the relevant results are in agreement with the results reported by Bhattacharyya et al. [27] with slight change of notations. The results obtained in particular cases are compared with the results presented by Bhattacharyya et al. [27]. Fig. 1(b) shows the excellent agreement, which justifies the accuracy of present numerical scheme. 4. Results and discussion In order to gain a clear insight of the physical problem, we have discussed the effects of different values of magnetic parameter M2 , mixed convection parameter k, velocity slip parameter (d), thermal slip parameter b, suction/blowing parameter S, Prandtl number Pr and Eckert number Ec and inclination c on the velocity, temperature, heat transfer rate and shear stress at the plate. M2 ¼ 0 represents the case when there is no applied magnetic field. The mixed convection parameter k represents a measure of the effect of the buoyancy in comparison with that of the inertia of the external forced or free stream flow on the heat and fluid flow. Forced convection is the dominant mode of transport when k ! 0, whereas free convection is the dominant mode when k ! 1. For the heated plate case ðTw > T1 Þ, the upward free convection flow caused by the buoyancy is in the same direction with the external forced convection flow. This case is called buoyancy aided mixed convection flow. For the cooling plate case ðTw < T1 Þ, the buoyancy causes a downward free convective flow which is in the opposite direction to that of upward external forced convection flow, which is called the buoyancy opposed mixed convection flow. The values of Pr are chosen 0.72 which represents air at 20 C temperature and 1 atmospheric pressure. Ec ¼ 0 corresponds to no Joule and viscous heating. The values of magnetic parameter and slip parameters are chosen arbitrarily.
Figure 3(b) Velocity profile for different d when M2 ¼ 0:5; Pr ¼ 0:72 and c ¼ p4 for buoyancy opposed flow (k < 0).
5 The fluid velocity fðgÞ0 increases with an increase in magnetic parameter M2 for both cases of buoyancy aided and opposed flows as shown in Fig. 2(a) and 2(b). An increase in the strength of magnetic parameter M2 , the Lorentz force associated with the magnetic field makes the boundary layer thinner. The magnetic lines of forces move past the plate at the free stream velocity. The fluid which is decelerated by the viscous force, receives a push from the magnetic field which counteracts the viscous effects. Hence the velocity of the fluid increases as the parameter M2 increases. Fig. 3(a) and 3(b) shows that the fluid velocity f 0 ðgÞ increases and as a result the momentum boundary layer thickness decreases with an increase in velocity slip parameter d for buoyancy aided/opposed flows. An increase in slip (in magnitude) more fluid get permits to slip past the plate and fluid experiences less drag and as a result, the flow accelerates near the plate and away from the plate this effect diminishes. There is a deceleration in the fluid velocity f 0 ðgÞ as the inclination c increases for both cases of buoyancy aided and opposed flows as shown in Fig. 4(a) and 4(b). The momentum boundary layer is found to be thickened for increasing values of c. For c ¼ p2, the plate is horizontal and for c ¼ 0 the plate assumes a vertical position. The gravitational effect is minimum for c ¼ p2 and maximum for c ¼ 0. The inclination parameter c arises only
Figure 4(a) Velocity profile for different c when Pr ¼ 0:72; M2 ¼ 0:5 and d ¼ 0:5 for buoyancy aided flow (k > 0).
Figure 4(b) Velocity profile for different c when Pr ¼ 0:72; M2 ¼ 0:5 and d ¼ 0:5 for buoyancy opposed flow(k < 0).
Please cite this article in press as: S. Das et al., Magnetohydrodynamic mixed convective slip flow over an inclined porous plate with viscous dissipation and Joule heating, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.03.003
6 in the buoyancy term k cos c in the momentum Eq. (11). Thus, the fluid velocity is found to be maximized at the vertical position of the plate (c ¼ 0) and minimized for the horizontal position of the plate (c ¼ p2). Fig. 5(a) and 5(b) displays the effects of Prandtl number on the fluid velocity f 0 ðgÞ. The fluid velocity f 0 ðgÞ decreases for increasing values of Prandtl number Pr for buoyancy aided/opposed flows. Physically speaking, the Prandtl number is an important parameter in heat transfer processes as it characterizes the ratio of thicknesses of the viscous and thermal boundary layers. Increasing the value of Pr causes the fluid temperature and its boundary layer thickness to decrease significantly as seen from Fig. 9(a) and 9(b). This decrease in temperature produces a net reduction of the thermal buoyancy effect in the momentum equation which results in less induced flow along the plate and consequently, the fluid velocity decreases. The momentum boundary layer thickness generally increases with increasing values of Pr. Physically, it is true as the Prandtl number describes the ratio between momentum diffusivity and thermal diffusivity and hence controls the relative thickness of the momentum and thermal boundary layers. As Pr increases the viscous forces (momentum diffusivity) dominate the thermal diffusivity and consequently decreases the velocity. Fig. 6 shows that the fluid velocity f 0 ðgÞ increases for increasing values of mixed convection parameter k for
Figure 5(a) Velocity profile for different Pr when M2 ¼ 0:5; d ¼ 0:5 and c ¼ p4 for buoyancy aided flow (k > 0).
Figure 5(b) Velocity profile for different Pr when M2 ¼ 0:5; d ¼ 0:5 and c ¼ p4 for buoyancy opposed flow (k < 0).
S. Das et al. suction/blowing. An increase at k in positive direction results in increasing the fluid velocity due to addition of buoyancy-induced flow onto the external forced convection flow. An increase at k in negative direction results in decreasing velocities due to retarding effect of downward buoyancy-induced flow onto the upward external forced convection flow. For the buoyancy-opposing case (k < 0), the velocity profiles are quite similar to the buoyancy-assisted case (k < 0), this is to say, the velocity profiles increase as k enlarges. The physical explanation is that in the buoyancy-opposing case, the buoyancy force plays a negative effect on the fluid motion in the boundary layer. Fig. 7 illustrates that the fluid velocity f 0 ðgÞ increases for increasing values of suction parameter S for both cases of buoyancy aided and opposed flows. Since the effect of suction is to suck away the fluid near the plate, the momentum boundary layer is reduced due to suction velocity. Consequently the velocity increases. Fig. 8(a) and 8(b) reveals that the fluid temperature hðgÞ decreases with an increase in magnetic parameter M2 for both cases of buoyancy aided and opposed flows. The thermal boundary layer thickness increases for increasing values of M2 . This can be attributed to the influence of Ohmic heating due to magnetic field in the flow. Fig. 9(a) and 9(b) represents the variation of fluid temperature hðgÞ with respect to Prandtl number Pr. The graph depicts that the fluid temperature decreases when the values of Prandtl number Pr increase for both cases of buoyancy aided and opposed flows. This is due to the fact that a higher Prandtl number fluid has relatively low thermal conductivity, which reduces conduction and thereby the thermal boundary layer thickness; and as a result, temperature decreases. Increasing Pr is to increase the heat transfer rate at the surface because the temperature gradient at the surface increases. Fig. 10(a) and 10(b) shows that the fluid temperature hðgÞ increases for increasing values of Eckert number Ec for buoyancy aided/opposed flows. The thermal boundary layer thickness decreases with increasing values of Ec. The viscous dissipation, as a heat generation inside the fluid, increases the bulk fluid temperature. This can be attributed to the additional heating in the flow system due to viscous dissipation. For the case of k ¼ 0, the Eckert number does not have any influence on temperature profile since the momentum and energy equations are not coupled. The fluid temperature hðgÞ decreases for increasing values of
Figure 6 Velocity profile for different k when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and c ¼ p4 in the presence of suction/injection.
Please cite this article in press as: S. Das et al., Magnetohydrodynamic mixed convective slip flow over an inclined porous plate with viscous dissipation and Joule heating, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.03.003
Magnetohydrodynamic mixed convective slip flow
7
Figure 7 Velocity profile for different S when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and c ¼ p4 for buoyancy aided/ opposed flow.
Figure 9(a) Temperature profiles for different Pr when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 for buoyancy aided flow (k > 0).
Figure 8(a) Temperature profiles for different M2 when Pr ¼ 0:72; d ¼ 0:5, b ¼ 0:2 and Ec ¼ 0:1 for buoyancy aided flow (k > 0).
Figure 9(b) Temperature profiles for different Pr when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 for buoyancy opposed flow (k < 0).
Figure 8(b) Temperature profiles for different M2 when Pr ¼ 0:72; d ¼ 0:5, b ¼ 0:2 and Ec ¼ 0:1 for buoyancy opposed flow (k < 0).
Figure 10(a) Temperature profiles for different Ec when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 for buoyancy aided flow (k > 0).
Please cite this article in press as: S. Das et al., Magnetohydrodynamic mixed convective slip flow over an inclined porous plate with viscous dissipation and Joule heating, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.03.003
8
S. Das et al.
Figure 10(b) Temperature profiles for different Ec when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 for buoyancy opposed flow (k < 0).
Figure 12(a) Temperature profiles for different b when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 for buoyancy aided flow (k > 0).
Figure 11(a) Temperature profiles for different d when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 for buoyancy aided flow (k > 0).
Figure 12(b) Temperature profiles for different b when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 for buoyancy opposed flow (k < 0).
Figure 11(b) Temperature profiles for different d when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 for buoyancy opposed flow (k < 0).
Figure 13 Temperature profiles for different k when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 in the presence of suction/injection.
Please cite this article in press as: S. Das et al., Magnetohydrodynamic mixed convective slip flow over an inclined porous plate with viscous dissipation and Joule heating, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.03.003
Magnetohydrodynamic mixed convective slip flow
9
Figure 14 Temperature profiles for different S when M2 ¼ 0:5; Pr ¼ 0:72, d ¼ 0:5 and Ec ¼ 0:1 for buoyancy aided/ opposed flow.
velocity slip parameter d for suction/blowing as shown in Fig. 11(a) and 11(b). The thermal boundary layer thickness increases with increasing values of d. Fig. 12(a) and 12(b) exhibits that the fluid temperature hðgÞ decreases for increasing values of thermal slip parameter b for both cases of buoyancy aided and opposed flows. As the thermal slip parameter b increases, less heat is transferred from the plate to the fluid and hence the temperature decreases for both buoyancy aided and opposed flows. The thermal boundary layer thickness becomes thicker for increasing values of b.
Table 1 The shear stress f 00 ð0Þ and the rate of heat transfer h0 ð0Þ at the plate g ¼ 0. M2
S
k
Pr
Ec
d
b
c
f 00 ð0Þ
h0 ð0Þ
0.5 1 2
0.5 0.5 0.5
0.2 0.2 0.2
0.72 0.72 0.72
0.1 0.1 0.1
0.5 0.5 0.5
0.2 0.2 0.2
p 4 p 4 p 4
0.82450 0.87451 0.95597
0.90292 0.89996 0.89573
0.5 0.5 0.5
0.5 0 0.5
0.72597 0.77541 0.82450
0.77433 0.83771 0.90292
0.77918 0.80187 0.82450
0.90121 0.90207 0.90292
0.82522 0.82484 0.82450
0.85728 0.88146 0.90292
0.82428 0.82450 0.82473
0.92146 0.90292 0.88438
0.82450 0.59019 0.45904
0.90292 0.91717 0.92393
0.82946 0.81975 0.81519
1.10717 0.70725 0.51961
0.83388 0.82450 0.80185
0.90326 0.90292 0.90207
0.2 0 0.2 0.25 0.5 0.72 0 0.1 0.2 0.5 1 1.5 0 0.5 1 0 p 4 p 2
Fig. 13 reveals that the fluid temperature hðgÞ decreases for increasing values of mixed convection parameter k for suction/blowing. Physically, in the process of cooling, the free convection currents are carried away from the plate to the free stream and since the free stream is in the upward direction and thus the free currents induce more fluid velocity to enhance. As a result, the thermal boundary layer thickness increases. Fig. 14 shows that the fluid temperature hðgÞ decreases for increasing values of suction parameter S for both cases of buoyancy aided and opposed flows. The explanation for such behavior is that the fluid is brought closer to the plate surface and reduces the thermal boundary layer thickness. As the distance x from the origin increases, the temperature of the plate decreases which is very clear from the expression of the plate temperature Tw ¼ T1 þ Tx0 as shown in temperature figures. For engineering purposes, one is usually interested in the values of the shear stress (skin friction) and the rate of heat transfer at the plate. The shear stress is an important parameter in the heat transfer studies, since it is directly related to the heat transfer coefficients. The increased shear stress is generally a disadvantage in the technical applications, while the increased heat transfer can be exploited in some applications such as heat exchangers, but should be avoided in other such as gas turbine applications, for instance. The numerical values of the rate of heat transfer h0 ð0Þ and the shear stress f 00 ð0Þ at the plate g ¼ 0 are entered in the Table 1 for several values of M2 ; S; k; Pr; Ec; d; b and c. It is seen from the Table 1 that the rate of heat transfer h0 ð0Þ decreases for increasing values of M2 ; Ec; b; c whereas it decreases for increasing values of S; k; Pr and d. This implies that an increase in Prandtl number is accompanied by an enhancement of the heat transfer rate at the surface of the sheet. The underlying physics behind this can be described as follows. When fluid attains a higher Prandtl number, its thermal conductivity is lowered down and so its heat conduction capacity diminishes. Thereby the thermal boundary layer thickness gets reduced. As a consequence, the heat transfer rate at the surface is increased. The increasing magnetic field also increases the thermal boundary layer thickness and as a result the dimensionless heat transfer rate decreases with an increase in the magnetic field. Buoyancy forces can enhance the rate of heat transfer at the plate when they assist the forced convection, and vice versa. The thermal boundary layer thickness decreases with the suction parameter S which causes an increase in the rate of heat transfer. The explanation for such behavior is that the fluid is brought closer to the surface and reduces the thermal boundary layer thickness. On the other hand, h0 ð0Þ < 0 means the heat transfer takes place from the surface to ambient fluid. The shear stress f 00 ð0Þ increases for increasing values of M2 ; S; k; Ec whereas it decreases for increasing values of Pr; d; b and c. 5. Conclusion In this paper, we examine the combined effects of Joule heating and viscous dissipation on an MHD mixed convective flow past an inclined porous plate. Using similarity transformation, the governing equations are transformed into self-similar equations. The numerical solutions of the self-similar equations are
Please cite this article in press as: S. Das et al., Magnetohydrodynamic mixed convective slip flow over an inclined porous plate with viscous dissipation and Joule heating, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.03.003
10 obtained using shooting technique with the help of forth order Runge–Kutta method. The effects of the pertinent parameters are discussed. Numerical results for temperature and velocity are presented graphically for pertinent parameters. Based on the obtained graphical and tabular results, the following conclusions can be summarized as follows: The fluid velocity and temperature enhance when the strength of magnetic field increases. The fluid velocity temperature accelerates due to increasing thermal buoyancy force. The fluid velocity reduces for increasing values of velocity slip parameter. An increase in velocity slip at the sheet leads to increase the fluid temperature within the boundary layer, while an increase in thermal slip reduces the temperature distribution. The fluid temperature inside the boundary layer reduces when the values of the Prandtl number increase. The slip parameters always lead to thinning of the thermal boundary layer. Joule heating and viscous dissipation have the effect to increase the fluid velocity, and accordingly decrease the rate of heat transfer and shear stress at the plate.
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Please cite this article in press as: S. Das et al., Magnetohydrodynamic mixed convective slip flow over an inclined porous plate with viscous dissipation and Joule heating, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.03.003